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derived in the
parent entry
and concerning numbers and elements aaa, bbb of an arbitrary
ring, may be generalised to the following

Theorem. If the sign of one factor in a ring product is changed, the sign of the product changes.

Corollary 1. The product of real numbers is equal to the product of their absolute values equipped with the “--” sign if the number of negative factors is odd and with “++” sign if it is even. Especially, any odd power of a negative real number is negative and any even power of it is positive.

Corollary 2. Let us consider natural powers of a ring element. If one changes the sign of the base, then an odd power changes its sign but an even power remains unchanged: